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With the exception of some modular forms, all the built-in
L-series have weakly multiplicative coefficients, so that
L(s)=∑an/ns with an=aman for m, n coprime.
For two such L-series, Magma allows the user to construct their
product and, provided that it makes sense, their quotient.
Poles: SeqEnum Default: []
Residues: SeqEnum Default: []
Precision: RngIntElt Default:
Let L1(s) and L2(s) be two L-series of the same weight
whose coefficients are weakly multiplicative, that is, they
satisfy an=aman for m, n coprime. This function constructs
their product L(s)=L1(s)L2(s).
If one of the L-series has zeros that cancel the poles of the
other L-series, the user should specify the list of poles for
L1^ * (s)L2^ * (s) using the Poles parameter and the
corresponding residues using the Residues parameter.
See Terminology for the terminology and Constructing a General L-Series
for the format of the poles and residues parameters.
The number of digits of precision to which the values L(s) are
to be computed may be specified using the Precision parameter.
If it is omitted, the precision is taken to be that of the default
real field.
Poles: SeqEnum Default: []
Residues: SeqEnum Default: []
Precision: RngIntElt Default:
Let L1(s) and L2(s) be two L-series whose coefficients an
are weakly multiplicative, that is, they satisfy an=aman for
m, n coprime. This function constructs
their quotient L(s)=L1(s)/L2(s).
This function assumes (but does not check!) that this quotient exists and
is a genuine L-function with finitely many poles.
If L2(s) happens to have zeros that give poles in the quotient,
the user must specify the list of poles of L1^ * (s)/L2^ * (s) using
the Poles parameter and the corresponding residues using the
Residues parameter.
See Terminology for the terminology and Constructing a General L-Series
for the format of Poles and Residues.
The number of digits of precision to which the values L(s) are
to be computed may be specified using the Precision parameter.
If it is omitted, the precision is taken to be that of the default
real field.
Precision: RngIntElt Default:
Sign: FldComElt Default:
Let L1 and L2 be L-functions such that L1(s)=L(V1, s) and
L2(s)=L(V2, s) are associated to systems of l-adic representations
V1 and V2 (`a la Serre). This function computes their tensor product
L(s)=L(V1 tensor V2, s). This can be used, for example, to twist
L-function by characters or higher-dimensional Artin representations
(see Examples Tensor Product of L-series Coming from l-adic Representations, Non-abelian Twist of an Elliptic Curve).
Note that, in particular, both L1(s) and L2(s) must have integer
conductor, weakly multiplicative coefficients and Hodge numbers in the
set {0, 1}. The argument ExcFactors is a list of tuples of
the form < p, v > or < p, v, Fp(x) > that give,
for each of the primes p where V1 and V2 both have bad reduction,
the valuation v of the conductor of V1 tensor V2 at p and the
inverse local factor at p. If the data is not provided for such a
prime p, Magma will attempt to compute the local factors by assuming
that the inertia invariants behave well at p,
(V1 tensor V2)> = V1> tensor V2>.
It will also compute the conductor exponents by predicting the tame and
wild degrees from the degrees of the local factors, but this does not
work if both V1 and V2 are wildly ramified at p.
The sign in the functional equation of L(V1 tensor V2, s) cannot be
determined from the signs of the factors, so it will be calculated
numerically from the functional equation. If the sign is known, the user
may specify it by means of the Sign parameter.
The number of digits of precision to which the values L(s) are
to be computed may be specified using the Precision parameter.
If it is omitted, the precision is taken to be that of the default
real field.
See Examples H108E10 and Tensor Product of L-series Coming from l-adic Representations, Non-abelian Twist of an Elliptic Curve.
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